Consider the problem 3.5! : I can understand the concept of having:
but in the real solution, it is actually:
Now can someone tell me why is the root of pi used and how did it originate in the use of finding non-integer factorials?
Consider the problem 3.5! : I can understand the concept of having:
but in the real solution, it is actually:
Now can someone tell me why is the root of pi used and how did it originate in the use of finding non-integer factorials?
This isn't correct, the definition n! = n*(n-1)! only holds if n is a natural number, this is the factorial.
It can be generalized to the "Gamma-function", defined for all real numbers except negative integers.
Your examaple (Gamma(1/2) = sqrt(pi)) can be elegantly showed if you know the relation between the Gamma and the Beta function.
I know that isn't correct it is just the use of pi that confuses me.
Why is it there? My main understanding of pi comes from the fact that it is how many times a circles diameter fits in it's circumference and I can't possibly see how the value of pi applies to this situation!
I just want to know if there is an explanation for Pi cropping up everywhere
Sorry if I don't seem grateful - I am very grateful! That "1+1/4+1/9+1/16+1/25+...+1/nē+..." you gave me is driving me insane and I fear that Pi may be my obsession for life now!
happens to be a very interesting number because it does appear in many strange cases all over mathematics. Usually infinite sums involve this number quite a lot.
The reason why .5! involves pi is because it is the way the extended definition of the factorial was defined. I never heard of anybody else being succesful in using a different definition for the extended factorial.